Quantum heat engines and refrigerators

File:Three-level-amp.pdf

The three-level-amplifier is the template of a quantum device. It operates by employing a hot and cold bath

to maintain population inversion between two energy levels which is used to amplify light by stimulated emissionYariv, Amnon (1989). Quantum Electronics, 3rd ed., Wiley. {{ISBN|0-471-60997-8}}

The ground state level (1-g) and the excited level (3-h) are coupled to a hot bath of temperature $T\_\backslash text\{h\}$.

The energy gap is $\backslash hbar\; \backslash omega\_\backslash text\{h\}\; =\; E\_3-E\_1$. When the population on the levels equilibrate

::$\backslash frac\{N\_\backslash text\{h\}\}\{N\_\backslash text\{g\}\}=e^\{-\backslash frac\{\backslash hbar\; \backslash omega\_\backslash text\{h\}\}\{k\_\backslash text\{B\}\; T\_\backslash text\{h\}\}\}$

where $\backslash hbar=\backslash frac\{h\}\{2\; \backslash pi\; \}$ is the Planck constant and $k\_\backslash text\{B\}$ is the Boltzmann constant.

The cold bath of temperature $T\_\backslash text\{c\}$ couples the ground (1-g) to an intermediate level (2-c)

with energy gap $E\_2-E\_1=\backslash hbar\; \backslash omega\_\backslash text\{c\}$.

When levels 2-c and 1-g equilibrate then

::$\backslash frac\{N\_\backslash text\{c\}\}\{N\_\backslash text\{g\}\}=e^\{-\backslash frac\{\backslash hbar\; \backslash omega\_\backslash text\{c\}\}\{k\_\backslash text\{B\}\; T\_\backslash text\{c\}\}\}$.

The device operates as an amplifier when levels (3-h) and (2-c) are coupled to an external field of frequency $\backslash nu$.

For optimal resonance conditions $\backslash nu=\backslash omega\_\backslash text\{h\}-\backslash omega\_\backslash text\{c\}$.

The efficiency of the amplifier in converting heat to power is the ratio of work output to heat input:

::$\backslash eta=\backslash frac\{\backslash hbar\; \backslash nu\}\{\backslash hbar\; \backslash omega\_\backslash text\{h\}\}\; =1-\backslash frac\{\backslash omega\_\backslash text\{c\}\}\{\backslash omega\_\backslash text\{h\}\}$.

Amplification of the field is possible only for positive gain (population inversion)

$G\; =N\_\backslash text\{h\}\; -\; N\_\backslash text\{c\}\; \backslash ge\; 0$.

This is equivalent to $\backslash frac\{\backslash hbar\; \backslash omega\_\backslash text\{c\}\}\{k\_\backslash text\{B\}\; T\_\backslash text\{c\}\}\; \backslash ge\; \backslash frac\{\backslash hbar\; \backslash omega\_\backslash text\{h\}\}\{k\_\backslash text\{B\}\; T\_\backslash text\{h\}\}$.

Inserting this expression into the efficiency formula leads to:

::$\backslash eta\; =1-\backslash frac\{\backslash omega\_\backslash text\{c\}\}\{\backslash omega\_\backslash text\{h\}\}\; \backslash le\; 1-\; \backslash frac\{T\_\backslash text\{c\}\}\{T\_\backslash text\{h\}\}\; =\; \backslash eta\_\backslash text\{c\}$

where $\backslash eta\_\backslash text\{c\}$ is the Carnot cycle efficiency.

Equality is obtained under a zero gain condition $G\; =0$.

The relation between the quantum amplifier and the Carnot efficiency was first pointed out by Scovil and Schultz-DuBois:

Reversing the operation driving heat from the cold bath to the hot bath by consuming power constitutes a refrigerator.

The efficiency of the refrigerator defined as the coefficient of performance (COP) for the reversed device is:

::$\backslash epsilon\; =\; \backslash frac\{\backslash omega\_\backslash text\{c\}\}\{\backslash nu\}\; \backslash le\; \backslash frac\{T\_\backslash text\{c\}\}\{T\_\backslash text\{h\}-T\_\backslash text\{c\}\}$

Quantum devices can operate either continuously or by a reciprocating cycle.

Continuous devices include solar cells converting solar radiation to electrical power, thermoelectric where the output is current and lasers where the output power is coherent light.

The primary example of a continuous refrigerator is optical pumping and laser cooling.{{cite journal|last1=Narevicius|first1=Edvardas|last2=Bannerman|first2=S Travis|last3=Raizen|first3=Mark G|title=Single-photon molecular cooling|journal=New Journal of Physics|volume=11|issue=5|year=2009|pages=055046|issn=1367-2630|doi=10.1088/1367-2630/11/5/055046|doi-access=free|bibcode=2009NJPh...11e5046N|arxiv=0808.1383}}{{cite journal|last1=Kosloff|first1=Ronnie|last2=Levy|first2=Amikam|title=Quantum Heat Engines and Refrigerators: Continuous Devices|journal=Annual Review of Physical Chemistry|volume=65|issue=1|year=2014|pages=365–393|issn=0066-426X|doi=10.1146/annurev-physchem-040513-103724|pmid=24689798|arxiv=1310.0683|bibcode=2014ARPC...65..365K|s2cid=25266545}}

Similarly to classical reciprocating engines, quantum heat engines also have a cycle that is divided into different strokes. A stroke is time segment in which a certain operation takes place (e.g. thermalization, or work extraction). Two adjacent strokes do not commute with each other. The most common reciprocating heat machines are the four-stroke machine, and the two-stroke machine. Reciprocating devices have been suggested operating either by the Carnot cycle{{cite journal|last1=Geva|first1=Eitan|last2=Kosloff|first2=Ronnie|title=A quantum-mechanical heat engine operating in finite time. A model consisting of spin-1/2 systems as the working fluid|journal=The Journal of Chemical Physics|volume=96|issue=4|year=1992|pages=3054–3067|issn=0021-9606|doi=10.1063/1.461951|bibcode=1992JChPh..96.3054G}}{{cite journal|last1=Bender|first1=Carl M|last2=Brody|first2=Dorje C|author-link2=Dorje C. Brody|last3=Meister|first3=Bernhard K|title=Quantum mechanical Carnot engine|journal=Journal of Physics A: Mathematical and General|volume=33|issue=24|year=2000|pages=4427–4436|issn=0305-4470|doi=10.1088/0305-4470/33/24/302|arxiv=quant-ph/0007002|bibcode=2000JPhA...33.4427B|s2cid=5335}} or the Otto cycle.{{cite journal|last1=Feldmann|first1=Tova|last2=Kosloff|first2=Ronnie|title=Performance of discrete heat engines and heat pumps in finite time|journal=Physical Review E|volume=61|issue=5|year=2000|pages=4774–4790|issn=1063-651X|doi=10.1103/PhysRevE.61.4774|pmid=11031518|bibcode=2000PhRvE..61.4774F|arxiv=physics/0003007|s2cid=2277942}}

In both types the quantum description allows to obtain equation of motion for the working medium

and the heat flow from the reservoirs.

{{see also|Reciprocating heat engine}}

Quantum versions of most of the common thermodynamic cycles have been studied, for example the Carnot cycle,{{cite journal|last1=Quan|first1=H. T.|last2=Liu|first2=Yu-xi|last3=Sun|first3=C. P.|last4=Nori|first4=Franco|title=Quantum thermodynamic cycles and quantum heat engines|journal=Physical Review E|volume=76|issue=3|pages=031105|year=2007|issn=1539-3755|doi=10.1103/PhysRevE.76.031105|pmid=17930197|bibcode=2007PhRvE..76c1105Q|arxiv=quant-ph/0611275|s2cid=3009953}} Stirling cycle{{cite journal|last1=Wu|first1=F.|last2=Chen|first2=L.|last3=Sun|first3=F.|last4=Wu|first4=C.|last5=Zhu|first5=Yonghong|title=Performance and optimization criteria for forward and reverse quantum Stirling cycles|journal=Energy Conversion and Management|volume=39|issue=8|year=1998|pages=733–739|issn=0196-8904|doi=10.1016/S0196-8904(97)10037-1|bibcode=1998ECM....39..733W }} and Otto cycle.{{cite journal|last1=Kieu|first1=T. D.|title=Quantum heat engines, the second law and Maxwell's daemon|journal=The European Physical Journal D|volume=39|issue=1|year=2006|pages=115–128|issn=1434-6060|doi=10.1140/epjd/e2006-00075-5|bibcode=2006EPJD...39..115K|arxiv=quant-ph/0311157|s2cid=119382163}}

The Otto cycle can serve as a template for other reciprocating cycles.

File:Q-otto-cycle.pdf shown in the Entropy $\backslash Omega$ plane

where the energy entropy and the Von Neumann entropy are displayed. $\backslash Omega$ is the

internal frequency of the device and is controlled externally. It mimics the inverse volume in the Otto cycle.

The red and blue lines are the hot and cold isochores. The cycle represents a heat pump.]]

It is composed of the following four segments:

• Segment $A\; \backslash rightarrow\; B$ isomagnetic or isochoric process, partial equilibration with the cold bath under constant Hamiltonian. The dynamics of the working medium is characterized by the propagator $\{\; U\}/$.
• Segment $B\; \backslash rightarrow\; C$ magnetization or adiabatic compression, the external field changes expanding the gap between energy levels of the Hamiltonian. The dynamics is characterized by the propagator $\{\; U\}\_\backslash text\{ch\}$.
• Segment $C\; \backslash rightarrow\; D$ isomagnetic, or isochoric process partial equilibration with the hot bath described by the propagator $U\_\backslash text\{h\}$.
• Segment $D\; \backslash rightarrow\; A$ demagnetization or adiabatic expansion reducing the energy gaps in the Hamiltonian, characterized by the propagator $U\_\backslash text\{hc\}$.

The propagator of the four stroke cycle

becomes $U\_\backslash text\{global\}$, which is the ordered product of the segment propagators:

::$$

{U}_\text{global}~~=~~ { U}_\text{hc} { U}_\text{h} {U}_\text{ch} {U}_\text{c}

The propagators are linear operators defined on a vector space which completely determines the state of the working medium.

Common to all thermodynamic cycles the consecutive segment propagators do not commute $[\{\backslash \; U\}\_i,\{\; U\}\_j]\; \backslash ne\; 0$.

Commuting propagators will lead to zero power.

In a reciprocating quantum heat engine the working medium is a quantum system such as spin systems{{cite journal|last1=Feldmann|first1=Tova|last2=Kosloff|first2=Ronnie|title=Quantum four-stroke heat engine: Thermodynamic observables in a model with intrinsic friction|journal=Physical Review E|volume=68|issue=1|pages=016101|year=2003|issn=1063-651X|doi=10.1103/PhysRevE.68.016101|pmid=12935194|bibcode=2003PhRvE..68a6101F|arxiv=quant-ph/0303046|s2cid=23777311}} or an harmonic oscillator.{{cite journal|last1=Rezek|first1=Yair|last2=Kosloff|first2=Ronnie|title=Irreversible performance of a quantum harmonic heat engine|journal=New Journal of Physics|volume=8|issue=5|year=2006|pages=83|issn=1367-2630|doi=10.1088/1367-2630/8/5/083|doi-access=free|bibcode=2006NJPh....8...83R|arxiv=quant-ph/0601006}} For maximum power the cycle time should be optimized.

There are two basic timescales in the reciprocating refrigerator the cycle time $\backslash tau\_\backslash text\{cyc\}$ and the internal

timescale $2\; \backslash pi\; /\backslash omega$. In general when $\backslash tau\_\backslash text\{cyc\}\; \backslash gg\; 2\backslash pi/\backslash omega$ the

engine operates in quasi-adiabatic conditions. The only quantum effect can be found at low temperatures

where the unit of energy of the device becomes $\backslash hbar\; \backslash omega$ instead of $k\_\backslash text\{B\}\; T$.

The efficiency at this limit is $\backslash eta\; =\; 1\; -\backslash frac\{\backslash omega\_\backslash text\{c\}\}\{\backslash omega\_\backslash text\{h\}\}$, always smaller than the Carnot efficiency $\backslash eta\_\backslash text\{c\}$. At high temperature and for the harmonic working medium the efficiency at maximum power becomes $\backslash eta\; =\; 1-\backslash sqrt\{\backslash frac\{T\_\backslash text\{c\}\}\{T\_\backslash text\{h\}\}\}$ which is the endoreversible thermodynamics result.

For shorter cycle times the working medium cannot follow adiabatically the change in the external parameter.

This leads to friction-like phenomena. Extra power is required to drive the system faster.

The signature of such dynamics is the development of coherence causing extra dissipation.

Surprisingly the dynamics leading to friction is quantized meaning that frictionless solutions to the adiabatic expansion/compression

can be found in finite time.{{cite journal|last1=Campo|first1=A. del|last2=Goold|first2=J.|last3=Paternostro|first3=M.|title=More bang for your buck: Super-adiabatic quantum engines|journal=Scientific Reports|volume=4|issue=1|pages=6208|year=2014|issn=2045-2322|doi=10.1038/srep06208|pmid=25163421|pmc=4147366|doi-access=free|arxiv=1305.3223 |bibcode=2014NatSR...4.6208C}}

{{cite journal|last1=Beau|first1=Mathieu|last2=Jaramillo|first2=Juan|last3=del Campo|first3=Adolfo|title=Scaling-Up Quantum Heat Engines Efficiently via Shortcuts to Adiabaticity|journal=Entropy|volume=18|issue=5|year=2016|pages=168|issn=1099-4300|doi=10.3390/e18050168|doi-access=free|bibcode=2016Entrp..18..168B|arxiv=1603.06019}} As a result, optimization has to be carried out only with respect to the time allocated

to heat transport. In this regime the quantum feature of coherence degrades the performance.

Optimal frictionless performance is obtained when the coherence can be cancelled.

The shortest cycle times $\backslash tau\_\backslash text\{cyc\}\; \backslash ll\; 2\backslash pi/\backslash omega$, sometimes termed sudden cycles,{{cite journal|last1=Feldmann|first1=Tova|last2=Kosloff|first2=Ronnie|title=Short time cycles of purely quantum refrigerators|journal=Physical Review E|volume=85|issue=5|pages=051114|year=2012|issn=1539-3755|doi=10.1103/PhysRevE.85.051114|pmid=23004710|bibcode=2012PhRvE..85e1114F|arxiv=1204.4059|s2cid=31174895}} have universal features. In this case coherence contributes to the cycles power.

A two-stroke engine quantum cycle equivalent to the Otto cycle based on two qubits has been proposed.

The first qubit has frequency $\backslash omega\_\backslash text\{h\}$ and the second $\backslash omega\_\backslash text\{c\}$. The cycle is composed

of a first stroke of partial equilibration of the two qubits with the hot and cold bath in parallel.

The second power stroke is composed of a partial or full swap between the qubits.

The swap operation is generated by a unitary transformation which preserves the entropy

as a result it is a pure power stroke.{{cite journal|last1=Allahverdyan|first1=Armen E.|last2=Hovhannisyan|first2=Karen|last3=Mahler|first3=Guenter|title=Optimal refrigerator|journal=Physical Review E|volume=81|issue=5|pages=051129|year=2010|issn=1539-3755|doi=10.1103/PhysRevE.81.051129|pmid=20866207|bibcode=2010PhRvE..81e1129A|arxiv=1007.4307|s2cid=12750223}}{{cite journal|last1=Uzdin|first1=Raam|last2=Kosloff|first2=Ronnie|title=The multilevel four-stroke swap engine and its environment|journal=New Journal of Physics|volume=16|issue=9|year=2014|pages=095003|issn=1367-2630|doi=10.1088/1367-2630/16/9/095003|doi-access=free|bibcode=2014NJPh...16i5003U|arxiv=1404.6182}}

The quantum Otto cycle refrigerators shares the same cycle with magnetic refrigeration.{{cite journal|last1=Shirron|first1=Peter J.|last2=McCammon|first2=Dan|title=Salt pill design and fabrication for adiabatic demagnetization refrigerators|journal=Cryogenics|volume=62|year=2014|pages=163–171|issn=0011-2275|doi=10.1016/j.cryogenics.2014.03.022|bibcode=2014Cryo...62..163S}}

Continuous quantum engines are the quantum analogues of turbines. The work output mechanism is coupling to an external periodic field, typically the electromagnetic field. Thus the heat engine is a model for a laser.

The models differ by the choice of their working substance

and heat source and sink. Externally driven two-level,{{cite journal|last1=Gelbwaser-Klimovsky|first1=D.|last2=Alicki|first2=R.|last3=Kurizki|first3=G.|title=Minimal universal quantum heat machine|journal=Physical Review E|volume=87|issue=1|pages=012140|year=2013|issn=1539-3755|doi=10.1103/PhysRevE.87.012140|pmid=23410316|bibcode=2013PhRvE..87a2140G|arxiv=1209.1190|s2cid=18826566}} three level{{cite journal|last1=Geva|first1=Eitan|last2=Kosloff|first2=Ronnie|title=The quantum heat engine and heat pump: An irreversible thermodynamic analysis of the three-level amplifier|journal=The Journal of Chemical Physics|volume=104|issue=19|year=1996|pages=7681–7699|issn=0021-9606|doi=10.1063/1.471453|bibcode=1996JChPh.104.7681G}} four-level{{cite journal|last1=Scully|first1=M. O.|last2=Chapin|first2=K. R.|last3=Dorfman|first3=K. E.|last4=Kim|first4=M. B.|last5=Svidzinsky|first5=A.|title=Quantum heat engine power can be increased by noise-induced coherence|journal=Proceedings of the National Academy of Sciences|volume=108|issue=37|year=2011|pages=15097–15100|issn=0027-8424|doi=10.1073/pnas.1110234108|pmid=21876187|pmc=3174605|doi-access=free|bibcode=2011PNAS..10815097S}}{{cite journal|last1=Harbola|first1=Upendra|last2=Rahav|first2=Saar|last3=Mukamel|first3=Shaul|title=Quantum heat engines: A thermodynamic analysis of power and efficiency|journal=EPL (Europhysics Letters)|volume=99|issue=5|year=2012|pages=50005|issn=0295-5075|doi=10.1209/0295-5075/99/50005|bibcode=2012EL.....9950005H|s2cid=13833767 |url=https://epljournal.edpsciences.org/10.1209/0295-5075/99/50005/pdf }} and coupled harmonic oscillators{{cite journal|last1=Kosloff|first1=Ronnie|title=A quantum mechanical open system as a model of a heat engine|journal=The Journal of Chemical Physics|volume=80|issue=4|year=1984|pages=1625–1631|issn=0021-9606|doi=10.1063/1.446862|bibcode=1984JChPh..80.1625K}} have been studied.

The periodic driving splits the energy level structure of the working medium. This splitting allows the two level engine to couple

selectively to the hot and cold baths and produce power. On the other hand, ignoring this splitting in the derivation of the equation of motion will violate the second law of thermodynamics.{{cite journal|last1=Szczygielski|first1=Krzysztof|last2=Gelbwaser-Klimovsky|first2=David|last3=Alicki|first3=Robert|title=Markovian master equation and thermodynamics of a two-level system in a strong laser field|journal=Physical Review E|volume=87|issue=1|pages=012120|year=2013|issn=1539-3755|doi=10.1103/PhysRevE.87.012120|pmid=23410296|bibcode=2013PhRvE..87a2120S|arxiv=1211.5665|s2cid=25511420}}

Non thermal fuels have been considered for quantum heat engines. The idea is to increase the energy content of the hot bath without

increasing its entropy. This can be achieved by employing coherence{{cite journal|last1=Scully|first1=M. O.|title=Extracting Work from a Single Heat Bath via Vanishing Quantum Coherence|journal=Science|volume=299|issue=5608|year=2003|pages=862–864|issn=0036-8075|doi=10.1126/science.1078955|pmid=12511655|bibcode=2003Sci...299..862S|s2cid=120884236|doi-access=free}} or a squeezed thermal bath.{{cite journal|last1=Roßnagel|first1=J.|last2=Abah|first2=O.|last3=Schmidt-Kaler|first3=F.|last4=Singer|first4=K.|last5=Lutz|first5=E.|title=Nanoscale Heat Engine Beyond the Carnot Limit|journal=Physical Review Letters|volume=112|issue=3|pages=030602|year=2014|issn=0031-9007|doi=10.1103/PhysRevLett.112.030602|pmid=24484127|bibcode=2014PhRvL.112c0602R|arxiv=1308.5935|s2cid=1826585}} These devices do not violate the second law of thermodynamics.

Two-stroke, Four-stroke, and continuous machine are very different from each other. However it was shown{{cite journal|last1=Uzdin|first1=Raam|last2=Levy|first2=Amikam|last3=Kosloff|first3=Ronnie|title=Equivalence of Quantum Heat Machines, and Quantum-Thermodynamic Signatures|journal=Physical Review X|volume=5|issue=3|pages=031044|year=2015|issn=2160-3308|doi=10.1103/PhysRevX.5.031044|doi-access=free|bibcode=2015PhRvX...5c1044U|arxiv=1502.06592}} that there is a quantum regime where all these machines become thermodynamically equivalent to each other. While the intra cycle dynamics in the equivalence regime is very different in different engine types, when the cycle is completed they all turn out to provide the same amount of work and consume the same amount of heat (hence they share the same efficiency as well). This equivalence is associated with a coherent work extraction mechanism and has no classical analogue. These quantum features have been demonstrated experimentally.{{cite journal|last1=Klatzow|first1=James|last2=Becker|first2=Jonas N |last3=Ledingham|first3=Patrick M|last4=Weinzetl|first4=Christian|last5=Kaczmarek|first5=Krzysztof T|last6=Saunders|first6=Dylan J|last7=Nunn|first7=Joshua|last8=Walmsley|first8=Ian A|last9=Uzdin|first9=Raam|last10=Poem|first10=Eilon|title=Experimental demonstration of quantum effects in the operation of microscopic heat engines|journal=Physical Review Letters|volume=122|issue=11|pages=110601|year=2019|issn=1079-7114|doi=10.1103/PhysRevLett.122.110601|pmid=30951320|bibcode=2019PhRvL.122k0601K|arxiv=1710.08716|s2cid=206318713}}

The elementary example operates under quasi equilibrium conditions. Its main quantum feature is the discrete energy level structure.

More realistic devices operate out of equilibrium possessing friction heat leaks and finite heat flow.

Quantum thermodynamics supplies a dynamical theory required for systems out of equilibrium such as heat engines, thus,

inserting dynamics into thermodynamics.

The theory of open quantum systems constitutes the basic theory. For heat engines a reduced description of the dynamics

of the working substance is sought, tracing out the hot and cold baths.

The starting point is the general Hamiltonian of the combined systems:

::$$

H = H_\text{s}s + H_\text{c} + H_\text{h}\text{h} + H_\text{sc}+H_\text{sh}

and the system Hamiltonian $H\_\backslash text\{s\}(t)$ is time dependent.

A reduced description leads to the equation of motion of the system:

::$$

\frac{d}{dt} \rho = -\frac{i}{\hbar}[H_\text{s},\rho] + L_\text{h} (\rho)+L_\text{c}(\rho)

where $\backslash rho$ is the density operator describing the state of the working medium and $L\_\backslash text\{h/c\}$ is the generator of dissipative dynamics

which includes the heat transport terms from the baths.

Using this construction, the total change in energy of the sub-system becomes:

::$$

\frac{d}{dt} E = \left\langle \frac{\partial H_\text{s}}{\partial t} \right\rangle + \langle L_\text{h} (H_\text{s}) \rangle+\langle L_\text{c} (H_\text{s}) \rangle

leading to the dynamical version of the first law of thermodynamics:

• The power $P\; =\backslash left\backslash langle\; \backslash frac\{\backslash partial\; H\}\{\backslash partial\; t\}\; \backslash right\backslash rangle$
• Heat currents $J\_\backslash text\{h\}\; =\; \backslash langle\; L\_\backslash text\{h\}\; (H\_\backslash text\{s\})\; \backslash rangle$ and $J\_\backslash text\{c\}\; =\; \backslash langle\; L\_\backslash text\{c\}\; (H\_\backslash text\{s\})\; \backslash rangle$.

The rate of entropy production becomes:

::$$

\frac{d S}{dt} = -\frac{J_\text{h}}{T_\text{h}}-\frac{J_\text{c}}{T_\text{c}} \ge 0

The global structure of quantum mechanics is reflected in the derivation of the reduced description.

A derivation which is consistent with the laws of thermodynamics is based on

the weak coupling limit.

A thermodynamical idealization assumes that the system and the baths are uncorrelated, meaning that the total state

of the combined system becomes a tensor product at all times:

::$$

\rho = \rho_\text{s} \otimes \rho_\text{h} \otimes \rho_\text{c} ~.

Under these conditions the dynamical equations of motion become:

\frac{d}{dt} \rho_\text{s} = {L} \rho_\text{s}~,

where $\{L\}$ is the Liouville superoperator described in terms of the system's Hilbert space,

where the reservoirs are described implicitly.

Within the formalism of quantum open system, $L$ can take the form of the

Gorini-Kossakowski-Sudarshan-Lindblad (GKS-L) Markovian generator or also known just as Lindblad equation

.{{cite journal|last1=Kosloff|first1=Ronnie|title=Quantum Thermodynamics: A Dynamical Viewpoint|journal=Entropy|volume=15|issue=12|year=2013|pages=2100–2128|issn=1099-4300|doi=10.3390/e15062100|doi-access=free|bibcode=2013Entrp..15.2100K|arxiv=1305.2268}} Theories beyond the weak coupling regime have been proposed.{{cite journal |last1=Ludovico |first1=M. F. |last2=Lim |first2=J. S. |last3=Moskalets |first3=M. |last4=Arrachea |first4=L. |last5=Sánchez |first5=D. |title=Dynamical energy transfer in ac-driven quantum systems |journal=Physical Review B |date=21 April 2014 |volume=89 |issue=16 |page=161306 |doi=10.1103/PhysRevB.89.161306 |bibcode=2014PhRvB..89p1306L |hdl=10261/116187 |s2cid=119265583 |hdl-access=free }}

{{cite journal|last1=Gallego|first1=R|last2=Riera|first2=A|last3=Eisert|first3=J|title=Thermal machines beyond the weak coupling regime|journal=New Journal of Physics|volume=16|issue=12|year=2014|pages=125009|issn=1367-2630|doi=10.1088/1367-2630/16/12/125009|doi-access=free|bibcode=2014NJPh...16l5009G|arxiv=1310.8349}}{{cite journal|last1=Esposito|first1=Massimiliano|last2=Ochoa|first2=Maicol A.|last3=Galperin|first3=Michael|title=Quantum Thermodynamics: A Nonequilibrium Green's Function Approach|journal=Physical Review Letters|volume=114|issue=8|pages=080602|year=2015|issn=0031-9007|doi=10.1103/PhysRevLett.114.080602|pmid=25768745|bibcode=2015PhRvL.114h0602E|arxiv=1411.1800|s2cid=11498686}}

The absorption refrigerator is of unique importance in setting an autonomous quantum device.

Such a device requires no external power and operates without external intervention in scheduling the operations

.{{cite journal|last1=Palao|first1=José P.|last2=Kosloff|first2=Ronnie|last3=Gordon|first3=Jeffrey M.|title=Quantum thermodynamic cooling cycle|journal=Physical Review E|volume=64|issue=5|pages=056130|year=2001|issn=1063-651X|doi=10.1103/PhysRevE.64.056130|pmid=11736037|bibcode=2001PhRvE..64e6130P|arxiv=quant-ph/0106048|s2cid=8201978}}{{cite journal|last1=Linden|first1=Noah|last2=Popescu|first2=Sandu|last3=Skrzypczyk|first3=Paul|title=How Small Can Thermal Machines Be? The Smallest Possible Refrigerator|journal=Physical Review Letters|volume=105|issue=13|pages=130401|year=2010|issn=0031-9007|doi=10.1103/PhysRevLett.105.130401|pmid=21230755|bibcode=2010PhRvL.105m0401L|arxiv=0908.2076|s2cid=2707740}}{{cite journal|last1=Levy|first1=Amikam|last2=Kosloff|first2=Ronnie|title=Quantum Absorption Refrigerator|journal=Physical Review Letters|volume=108|issue=7|pages=070604|year=2012|issn=0031-9007|doi=10.1103/PhysRevLett.108.070604|pmid=22401189|bibcode=2012PhRvL.108g0604L|arxiv=1109.0728|s2cid=6981288}} The basic construct includes three baths; a power bath, a hot bath and a cold bath.

The tricycle model is the template for the absorption refrigerator.

File:Tricyle-1.png

The tricycle engine has a generic structure.

The basic model consists of three thermal baths: A hot bath with temperature $T\_\backslash text\{h\}$,

a cold bath with temperature $T\_\backslash text\{c\}$

and a work bath with temperature $T\_\backslash text\{d\}$.

Each bath is connected to the engine via a frequency filter which can be modeled by three oscillators:

::$$

H_0 = \hbar \omega_\text{h} a^{\dagger} a +\hbar \omega_\text{c} b^{\dagger} b + \hbar \omega_\text{d} c^{\dagger} c~~,

where $\backslash omega\_\backslash text\{h\}$, $\backslash omega\_\backslash text\{c\}$ and $\backslash omega\_\backslash text\{d\}$

are the filter frequencies on resonance $\backslash omega\_\backslash text\{d\}=\backslash omega\_\backslash text\{h\}-\backslash omega\_\backslash text\{c\}$.

The device operates as a refrigerator by removing an excitation from the cold bath as well as from the work bath

and generating an excitation in the hot bath. The term $a^\{\backslash dagger\}\; b\; c$ in the Hamiltonian is non linear

and crucial for an engine or a refrigerator.

::$$

H_I = \hbar \epsilon \left( a b^{\dagger} c^{\dagger} + a^{\dagger} b c \right)~~,

where $\backslash epsilon$ is the coupling strength.

The first-law of thermodynamics represents the energy balance of heat currents originating from the three baths and collimating on the system:

::$$

\frac{dE_\text{s}}{dt}= { J}_\text{h} + { J}_\text{c} +{J}_\text{d} ~~.

At steady state no heat is accumulated in the tricycle, thus $\backslash frac\{dE\_\backslash text\{s\}\}\{dt\}=\; 0$.

In addition, in steady state the entropy is only generated in the baths, leading to the second law of thermodynamics:

::$$

\frac{d}{dt}\Delta { S}_\text{u}~=~-\frac{{J}_\text{h}}{T_\text{h}} - \frac{{ J}_\text{c}}{T_\text{c}} -\frac{{ J}_\text{d} }{T_\text{d}}~\ge~0~~.

This version of the second-law is a generalisation of the statement of Clausius theorem;

heat does not flow spontaneously from cold to hot bodies.

When the temperature $T\_\backslash text\{d\}\; \backslash rightarrow\; \backslash infty$, no entropy is generated in the power bath.

An energy current with no accompanying entropy production is equivalent to generating pure power:

$\{\; P\}=\{J\}\_\backslash text\{d\}$, where $\{\; P\}$ is the power output.

There are seemingly two independent formulations of the third law of thermodynamics both originally were stated by Walther Nernst. The first formulation is known as the Nernst heat theorem, and can be phrased as:

• The entropy of any pure substance in thermodynamic equilibrium approaches zero as the temperature approaches zero.

The second formulation is dynamical, known as the unattainability principle:{{cite journal|last1=Landsberg|first1=P. T.|title=Foundations of Thermodynamics|journal=Reviews of Modern Physics|volume=28|issue=4|year=1956|pages=363–392|issn=0034-6861|doi=10.1103/RevModPhys.28.363|bibcode=1956RvMP...28..363L}}

• It is impossible by any procedure, no matter how idealized, to reduce any assembly to absolute zero temperature in a finite number of operations.

At steady state the second law of thermodynamics implies that the total entropy production is non-negative.

When the cold bath approaches the absolute zero temperature,

it is necessary to eliminate the entropy production divergence at the cold side

when $T\_\backslash text\{c\}\; \backslash rightarrow\; 0$, therefore

::$$

\dot S_\text{c} \propto - T_\text{c}^{\alpha}~~~,~~~~\alpha \geq 0~~.

For $\backslash alpha=0$ the fulfillment of the second law depends on the entropy production of the other baths,

which should compensate for the negative entropy production of the cold bath.

The first formulation of the third law modifies this restriction.

Instead of $\backslash alpha\; \backslash geq\; 0$ the third law imposes $\backslash alpha\; >\; 0$,

guaranteeing that at absolute zero the entropy production at the cold bath is zero: $\backslash dot\; S\_\backslash text\{c\}\; =\; 0$.

This requirement leads to the scaling condition of the heat current $\{\; J\}\_\backslash text\{c\}\; \backslash propto\; T\_\backslash text\{c\}^\{\backslash alpha+1\}$.

The second formulation, known as the unattainability principle can be rephrased as;{{cite journal|last1=Levy|first1=Amikam|last2=Alicki|first2=Robert|last3=Kosloff|first3=Ronnie|title=Quantum refrigerators and the third law of thermodynamics|journal=Physical Review E|volume=85|issue=6|pages=061126|year=2012|issn=1539-3755|doi=10.1103/PhysRevE.85.061126|pmid=23005070|bibcode=2012PhRvE..85f1126L|arxiv=1205.1347|s2cid=24251763}}

• No refrigerator can cool a system to absolute zero temperature at finite time.

The dynamics of the cooling process is governed by the equation

::$$

{ J}_\text{c}(T_\text{c}(t)) = -c_V(T_\text{c}(t))\frac{dT_\text{c}(t)}{dt}~~.

where $c\_V(T\_\backslash text\{c\})$ is the heat capacity of the bath. Taking $\{\; J\}\_\backslash text\{c\}\; \backslash propto\; T\_\backslash text\{c\}^\{\backslash alpha+1\}$ and $c\_V\; \backslash sim\; T\_\backslash text\{c\}^\{\backslash eta\}$ with $\{\backslash eta\}\; \backslash geq\; 0$, we can quantify this formulation by evaluating the characteristic exponent $\backslash zeta$ of the cooling process,

::$$

\frac{dT_\text{c}(t)}{dt} \propto -T_\text{c}^{\zeta}, ~~~~~ T_\text{c}\rightarrow 0, ~~~~~ {\zeta=\alpha-\eta+1}

This equation introduce the relation between the characteristic exponents $\backslash zeta$ and $\backslash alpha$. When then the bath is cooled to zero temperature in a finite time, which implies a violation of the third law. It is apparent from the last equation, that the unattainability principle is more restrictive than the Nernst heat theorem.

{{Reflist}}

Deffner, Sebastian and Campbell, Steve. "Quantum Thermodynamics: An introduction to the thermodynamics of quantum information", (Morgan & Claypool Publishers, 2019).{{Cite book|title=Quantum Thermodynamics|doi=10.1088/2053-2571/ab21c6|year=2019|last1=Deffner|first1=Sebastian|isbn=978-1-64327-658-8|s2cid=195791624}}

F. Binder, L. A. Correa, C. Gogolin, J. Anders, G. Adesso (eds.) "Thermodynamics in the Quantum Regime. Fundamental Aspects and New Directions." (Springer 2018)

Gemmer, Jochen, M. Michel, and Günter Mahler. "Quantum thermodynamics. Emergence of thermodynamic behavior within composite quantum systems. 2." (2009).

Petruccione, Francesco, and Heinz-Peter Breuer. The theory of open quantum systems. Oxford university press, 2002.

• {{cite web | url=http://phys.org/news/2014-01-nanoscale-standard-efficiency-limit.html | title=Nanoscale heat engine exceeds standard efficiency limit | website=phys.org}}

Category:Quantum mechanics

Category:Heat pumps

Category:Thermodynamics